First-neighbor distance

The definition of the first and second peaks is preserved reasonably well with increasing temperature and pressure up to about 1300 °C. However, above this temperature, both peaks broaden dramatically. The width of the first peak at the highest temperature suggests a distribution of first neighbor distances extending over at least 0.5 A (twice the low-temperature value) and the second peak is broadened to near invisibility. 

The a(t) depends nonhnearly on temperature in the low-temperature range, and the a(t) decreases with the feature sizes of nanostructures [8,9]. EXAFS investigations [10] revealed that in small gold particles, the temperature dependence of the first neighbor distance is different from that of the macrocrystalline counterpart. In the largest size samples, a reduction in the thermal expansion happens, whereas in the smallest ones, a crossover from an initial thermal expansion to a thermal contraction presents. 

Since the mean OOO angle 0 —109.5° in ice Ih is determined by the distance ratio of the peaks descriptive of first and second neighbor distributions, one can estimate its rms-variation, <(/10)2>. If it is assumed that the variance in this angle is given by the variance in the second neighbor distance alone, its magnitude should be <(/40)2> =0.166/2.751 =0.06 radians, or about 3°. 

Correlation functions hoo(R) for the amorphous deposit prepared and studied at 77 K are shown in Fig. 7b together with the curve for polycrystalline ice Ih. As in the crystalline phase, the nearest-neighbor oxygen-oxygen correlations in H20(as) occur in an exceptionally narrow band centered at 2.76 A, with rms-deviation 0.114 A. The distance ratio for second and first neighbors indicates tetrahedral coordination on the average, but the second neighbor peak near... 

The underlying metal lattice structure also has a significant effect on the water structure. As pointed out by Spohr," although the Pt-Pt nearest-neighbor distance is ao/ Jl = 0.277 nm, which is very close to the 0-0 distance in ice, the cubic symmetry of the 100 surface is incompatible with the hexagonal symmetry of the ice lattice. As a result, the water molecules cannot form a uniform monolayer and occupy all adsorption sites. On the other hand, Berkowitz and co-workers showed that the hexagonal Pt (111) surface is able to support a more complete layer of adsorbed water molecules, and one can identify patches of an icelike structure in the first layer. This freezing is further enhanced by an external electric field, as will be discussed later. 

To approach the complexity of a real tridimensional structure, let us first consider the case of a monodimensional array of alternating positive and negative charges, each at distance r from its first neighbor. We will assume for the sake of simplicity that the dispersive potential is negligible. The total potential is therefore... 

Consider a diatomic chain in which the atoms, of distinct masses mi and m2, are positioned at distance a (figure 3.6). The repetition distance of the chain is 2a, and the Brillouin zone falls between —irlla and i lla. If only the interactions between first neighbors are significant, the equation of motion for atom r at position ja is given by... 

Solution. There are six first-neighbor sites in the primitive cubic lattice, and the first-neighbor jump distance, r, is equal to the lattice constant, a. Once an atom has jumped into a given site, the probability that it will next jump into any of its nearest-neighbor sites (with the exception of the site from which it just jumped) is (1 —p)/5. Therefore,... 

Since swelling introduces an affine deformation of the network, the above calculation holds for any swelling degree and the distance d between first neighbor crosslinks can be written as ... 

This expression can be used to evaluate the intercrosslink distance dc in the nascent gel, whereby Qc = t "1 is the swelling degree of the gel upon its formation. However the intercrosslink distance d can only be considered identical with (r2)1/2 - the root mean square end-to-end distance of the elastic chains - if the overwhelming majority of the elastic chains connect first neighbor crosslinks, i.e., when the elastic chains hardly interpenetrate. 

The most important result obtained on the structure of model networks is the existence of a rather well-defined correlation distance between first neighbor crosslinks. The affine character of deformations induced by swelling processes is clearly shown by the fact that, for a given gel, the correlation distance d increases as the cubic root of the volume swelling degree Q up to values of Q of the order of 10. 

Elastic neutron diffraction was first performed (analyzer in Fig. 1 set to zero energy transfer) to establish the structure of the monolayer at low temperature. Three Bragg reflections were observed which could be indexed by a triangular lattice having a nearest-neighbor distance about 10% smaller than required for a 3 X /3 R30° commensurate structure (every third carbon hexagon in the graphite basal plane occupied). 

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